Problems

During the ball every young man danced the waltz with a girl, who was either more beautiful than the one he danced with during the previous dance, or more intelligent, but most of the men (at least 80%) – with a girl who was at the same time more beautiful and more intelligent. Could this happen? (There was an equal number of boys and girls at the ball.)

On a plane there is a square, and invisible ink is dotted at a point $P$. A person with special glasses can see the spot. If we draw a straight line, then the person will answer the question of on which side of the line does $P$ lie (if $P$ lies on the line, then he says that $P$ lies on the line).

What is the smallest number of such questions you need to ask to find out if the point $P$ is inside the square?

Ten little circles are drawn on a squared board $4\times 4$.

Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.

Cut a square into a heptagon (7 sides) and an octagon (8 sides) in such a way, that for every side of an octagon there exists an equal side belonging to the heptagon.

Cut a rectangle into two identical pentagons.

(a) Cut the rectangle into two identical quadrilaterals.

(b) Cut the rectangle into two identical hexagons.

(c) Cut the rectangle into two identical heptagons.

Can one cut a square into (a) one 30-gon and five pentagons? (b) one 33-gon and three 10-gons?

How can you divide a pancake with three straight sections into 4, 5, 6, 7 parts?

What is the maximum number of pieces that a round pancake can be divided into with three straight cuts?

In Neverland, there are magic laws of nature, one of which reads: “A magic carpet will fly only when it has a rectangular shape.” Frosty the Snowman had a magic carpet measuring $9\times 12$. One day, the Grinch crept up and cut off a small rug of size $1\times 8$ from this carpet. Frosty was very upset and wanted to cut off another $1\times 4$ piece to make a rectangle of $8\times 12$, but the Wise Owl suggested that he act differently. Instead he cut the carpet into three parts, of which a square magic carpet with a size of $10\times 10$ could be sown with magic threads. Can you guess how the Wise Owl restructured the ruined carpet?